where ii is monic. (This implies in particular that TT preserves monos.)

Definition

Let 𝒟\mathcal{D} be a category with pullbacks, and let η:S→T\eta: S \to T be a natural transformation between functors S,T:𝒞→𝒟S, T: \mathcal{C} \to \mathcal{D}. Then η\eta is taut if the naturality square

is a pullback. By hypothesis, the canonical map ϕ:T(A)→T(A)×T(B)T(A)\phi: T(A) \to T(A) \times_{T(B)} T(A) is (regular) epic, but it is also monic because its composition with either projection T(A)×T(B)T(A)→T(A)T(A) \times_{T(B)} T(A) \to T(A) is the identity. Therefore ϕ\phi is an isomorphism, i.e., applying TT to the displayed pullback is a pullback, and this forces T(i)T(i) to be monic.

is a pullback with ii monic, we have that jj and therefore T(j)T(j) is monic. The canonical map ϕ:T(P)→T(A)×T(C)T(B)\phi: T(P) \to T(A) \times_{T(C)} T(B) is (regular) epic, but also monic, since the mono T(j)T(j) factors through it. Thus ϕ\phi is an isomorphism, which completes the proof.

A similar proof shows that weakly cartesian natural transformations are also taut.

The covariant power set monad, whose algebras are sup-lattices, is taut.

An analytic endofunctor induced by a species is taut. Furthermore, a morphism of species induces a weakly cartesian transformation between the corresponding analytic functors, thus a fortiori a taut transformation. In particular, an analytic monad is taut.

As an exception, we have

The double (contravariant) power set functor P∘Pop:Set→SetP \circ P^{op}: Set \to Set is not taut.

Applications

Paul Taylor has made tautness of TT a central assumption in his account of induction via well-founded coalgebras over TT. See chapter VI of his book.

Tautness assumptions play a role in viewing relational TT-algebras and related structures as generalized multicategories in the sense of Cruttwell-Shulman. In the prototypical case of relational beta-modules, there is a virtual double category of relations. A taut monad TT on SetSet (such as the ultrafilter monad) induces a monad T¯\bar{T} on this virtual double category (that is, a monad in an appropriate 2-category of virtual double categories). From there, one can define a horizontal Kleisli construction which is another virtual double category HKl(Rel,T¯)HKl(Rel, \bar{T}), and a T¯\bar{T}-multicategory in RelRel is by definition a monoid in HKl(Rel,T¯)HKl(Rel, \bar{T}). In the special case T=βT = \beta, the ultrafilter monad, this concept recapitulates Barr’s notion of relational β\beta-module as synonym of “topological space”. This can be generalized further by working with a virtual double category of “VV-matrices” where VV is a completely distributive quantale (RelRel being the case V=2V = \mathbf{2}). Again with TT a taut monad, one can define a virtual double category HKl(V-Mat,T¯)HKl(V\text{-}Mat, \bar{T}) and then define generalized multicategories as before. (These were studied in a series of articles by Clementino, Hofmann, Tholen, Seal and others under the name “(T,V)(T, V)-algebras”.)